40 DYNAMICAL THEOKY OF SOUND 



of the corresponding frequency ; this case has already been 

 referred to in 13. In the second mode the ratio A : B is small, 

 as appears from the second of equations (17); M is then nearly 

 at rest, whilst m oscillates like the bob of a pendulum of 

 length b. 



Since the expression on the left-hand side of (20) cannot 

 vanish, the two frequencies can never exactly coincide, but they 

 become approximately equal if a = 6, nearly, and //, is small. 

 A curious phenomenon may then present itself. The motion 

 of each mass, being made up of two superposed simple-harmonic 

 vibrations of nearly equal period, may fluctuate greatly in 

 extent, and if the amplitudes of the two vibrations are equal 

 we have periods of approximate rest, as explained in 10. The 

 motion then appears to be transferred alternately from m to M, 

 and from M to m, at regular intervals*. 



If, on the other hand, M is small compared with m, p is nearly 

 equal to unity, and the two roots of (19) are ?i 2 = g/(a + b) and 

 n* = mg/M . (a -f b)/ab, approximately. The former root makes 

 B/A = (a + b)/a, nearly, so that the two masses are always 

 nearly in a line with the point of suspension, m now oscillating 

 like the bob of a pendulum of length a + b. In the second 

 mode the ratio B/A is small, so that m is approximately at 

 rest ; the motion of M is then like that of a particle attached 

 to a string which is stretched between fixed points with a 

 tension mg (cf. 6). 



Another case of interest is obtained if we make a infinite. 

 One root of (19) then vanishes, and the other is 



which makes A/B = - m/M. This indicates that if the support 

 of a simple pendulum yield horizontally, but without elasticity, 

 the frequency is increased in a certain ratio which is of course 



* The influence of dissipation is of course here neglected. If m be subject 

 to a frictional resistance, and especially if the modulus of decay be less than 

 the period of the fluctuation given by the above theory, the phenomena are 

 modified, and the illustration of the theory of resonance ( 12) is improved. 

 There is now a continual, though possibly a slow, drain on the original energy 

 otM. 



